= Saturation of a subset
{title2=$\operatorname{Sat}(A)=p^{-1}(p(A))$}
For the <canonical projection of a quotient set> $p:X\to X/{\sim}$, the saturation of $A\subseteq X$ is the union of all <equivalence classes> meeting $A$, namely $p^{-1}(p(A))$. A subset is called saturated when it equals its saturation. The <quotient topology> says that $p(A)$ is <closed> exactly when the saturation of $A$ is <closed>, and similarly for openness. A <closed set> need not have <closed> saturation: collapsing $(0,1)\subset\mathbb R$ to one point makes the saturation of $\{1/2\}$ equal to $(0,1)$. Hence a <quotient map> need not be a <closed map>.
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